Scale symmetry is notoriously fickle: even when (approximately) present at the classical level, quantum fluctuations often break it, sometimes rather dramatically. Indeed, contemporary physics encompasses the study of very different phenomena at very different scales, e.g., from the (nominally) meV scale of spin systems, via the eV of electronic band structures, to the GeV of elementary particles, and possibly even the 10¹⁹ GeV of quantum gravity. However, there are often – possibly surprising – analogies between systems across these seemingly disparate settings. Studying the possible emergence of quantum scale symmetry and its breakdown is one way to systematically exploit these similarities, and in fact allows one to make testable predictions within a unified technical framework (viz., the renormalization group). The aim of this thesis is to do so for a few explicit scenarios. In the first four of these, quantum scale symmetry emerges in the long-wavelength limit near a quantum phase transition, over length scales of the order of the correlation length. In the fifth example, quantum scale symmetry is restored at very high energies (i.e., at and above the Planck scale), but severely constrains the phenomenology at 'low' energies (e.g., at accelerator scales), despite scale invariance being badly broken there. We begin with the Gross–Neveu (= chiral) SO(3) transition in D = 2+1 spacetime dimensions, which notably has been proposed to describe the transition of certain spin-orbital liquids to antiferromagnets. The chiral fermions that suffer a spontaneous breakdown of their isospin symmetry in this setting are fractionalized excitations (called spinons), and are as such difficult to observe directly in experiment. However, as gapless degrees of freedom, they leave their imprint on critical exponents, which may hence serve as a diagnostic tool for such unconventional excitations. These may be computed using (comparatively) conventional field-theoretic techniques. Here, we employ three complementary methods: a three-loop expansion in D = 4 - ε spacetime dimensions, a second-next-leading order expansion in large flavour number N , and a non-perturbative calculation using the functional renormalization group in the improved local potential approximation. The results are in fair agreement with each other, and yield combined best-guess estimates that may serve as benchmarks for numerical simulations, and possibly experiments on candidate spin liquids. We next turn our attention to spontaneous symmetry breaking at zero temperature in quasi-planar (electronic) semimetals. We begin with Luttinger semimetals, i.e., semimetals where two bands touch quadratically at isolated points of the Brioullin zone; Bernal-stacked bilayer graphene (BBLG) within certain approximations is one example. Luttinger semimetals are unstable at infinitesimal 4-Fermi interaction towards an ordered state (i.e., the field theory is asymptotically free rather than safe). Nevertheless, since the interactions are marginal, there are several pathologies in the critical behaviour. We show how these pathologies may be understood as a collision between the IR-stable Gaußian fixed point and a critical fixed point distinct from the Gaußian one in d = 2 + ε spatial dimensions. Observables like the order-parameter expectation value develop essential rather than power-law singularities; their exponent, as shown herein by explicit computation for the minimal model of two-component ‘spinors’, is distinct from the mean-field one. More tellingly, although finite critical exponents often default to canonical power-counting values, the susceptibility exponent turns out to be one-loop exact, and, in said minimal model takes the value γ = 2γᵐᵉᵃⁿ⁻ᶠᶦᵉˡᵈ = 2. Such an exact yet non-mean-field prediction can serve as a useful benchmark for numerical methods. We then proceed to scenarios in D = 2 + 1 spacetime dimensions where Dirac fermions can arise from Luttinger fermions due to low rotational symmetry. In BBLG, the 'Dirac from Luttinger' mechanism can occur both due to explicit and spontaneous breaking of rotational symmetry. The explicit symmetry breaking is due to the underlying honeycomb lattice, which only has C₃ symmetry around the location of the band crossings (so-called K points). As a consequence, the quadratic band crossing points each split into four Dirac cones, which is shown explicitly by computing the two-loop self-energy in the 4-Fermi theory. Within our approximations, we can estimate the critical coupling up to which a semimetallic state survives; it is finite (unlike a quadratic band touching point with high rotational symmetry), but significantly smaller than a 'vanilla' Dirac semimetal. Based on the ordering temperature of BBLG, our rough estimate further shows that the (effective) coupling strength in BBLG may be close to the critical value, in sharp contrast to other quasi-planar Dirac semimetals (such as monolayer graphene). Rotational symmetry in BBLG may also be broken spontaneously, i.e., due to the presence of nematic order, whereby a quadratic band crossing splits into two Dirac cones. Such a scenario is also very appealing for BBLG, since the precise nature of the ordered ground state of BBLG has not been established unambiguously: whilst some experiments show an insulating ground state with a full bulk gap, others show a partial gap opening with four isolated linear band crossings. Here, we show within a simplified phenomenological model using mean-field theory that there exists an extended region of parameter space with coexisting nematic and layer-polarized antiferromagnetic order, with a gapless nematic phase on one side and a gapped antiferromagnetic phase on the other. We then show that the nematic-to-coexistence quantum phase transition has emergent Lorentz invariance to one-loop in D = 2 + ε as well as D = 4 - ϵ dimensions, and thus falls into the celebrated Gross-Neveu-Heisenberg universality class. Combining previous higher-order field-theoretic results, we derive best-guess estimates for the critical exponents of this transition, with the theoretical uncertainty coming out somewhat smaller than in the monolayer counterpart due to the enlarged number of fermion components. Overall, BBLG may hence be a promising candidate for experimentally accessible Gross–Neveu quantum criticality in D = 2 + 1 spacetime dimensions. Finally, we turn our attention to the 'low-energy' consequences of transplanckian quantum scale symmetry. Extensions to the Standard Model that tend to lower the Higgs mass have many phenomenologically attractive properties (e.g., it would allow one to accommodate a more stable electroweak vacuum). Dark matter is one well-motivated candidate for such an extension. However, even in the most conservative settings, one usually has to contend with a significantly enlarged number of free parameters, and a concomitant reduction of predictivity. Here, we investigate how asymptotic safety (i.e., imposing quantum scale symmetry at the Planck scale and above) may constrain the Higgs mass in Standard Model (plus quantum gravity) when coupled to Yukawa dark matter via a Higgs portal. Working in a toy version of the Standard Model consisting of the top quark and the radial mode of the Higgs, we show within certain approximations that the Higgs mass may be lowered by the necessary amount if the dark scalar undergoes spontaneous symmetry breaking, as a function of the dark scalar mass, which is the only free parameter left in the theory.:1 Introduction 1.1 Scale invariance – why and where 1.1.1 Fundamental quantum field theories 1.1.2 Universality 1.1.3 Novel phases of matter 1.2 Outline of this thesis 2 Renormalization Group: A Brief Review 2.1 Quantum fluctuations and generating functionals 2.2 Renormalization group flow 2.3 Basic notions 2.4 Scale transformations, scale symmetry and RG fixed points 2.5 Characterization and interpretation of RG fixed points 2.5.1 Formal aspects 2.5.2 Scaling at (quantum) phase transitions 2.5.3 Predictivity in fundamental physics 2.5.4 Effective asymptotic safety in particle physics and condensed matter 3 Gross–Neveu SO(3) Quantum Criticality in 2 + 1 Dimensions 3.1 Effective field theory 3.2 Renormalization and critical exponents 3.2.1 4 - ϵ expansion 3.2.1.1 Method 3.2.1.2 Flow equations 3.2.1.3 Critical exponents 3.2.2 Large-N expansion 3.2.2.1 Method 3.2.2.2 Critical exponents 3.2.3 Non-perturbative FRG 3.2.3.1 Flow equations 3.2.3.2 Representation of the effective potential 3.2.3.3 Choice of regulator 3.2.3.4 Limiting behaviour 3.3 Discussion 3.3.1 General behaviour and qualitative aspects 3.3.2 Quantitative estimates for D = 3 3.4 Summary and outlook 4 Luttinger Fermions in Two Spatial Dimensions 4.1 Introduction 4.2 Action from top-down construction 4.3 Renormalization 4.3.1 4-Fermi formulation 4.3.2 Yukawa formulation 4.4 Fixed-point analysis 4.5 Non-mean-field behaviour 4.5.1 Order-parameter expectation value 4.5.2 Susceptibility exponent 4.6 Bottom-up construction: Spinless fermions on kagome lattice 4.6.1 Tight-binding dispersion 4.6.2 From Hubbard to Fermi 4.6.3 Fate of particle-hole asymmetry 4.7 Discussion 5 Dirac from Luttinger I: Explicit Symmetry Breaking 5.1 From lattice to continuum 5.1.1 Fermions on Bernal-stacked honeycomb bilayer 5.1.2 Continuum limit 5.1.3 Interactions 5.2 Mean-field theory 5.3 Renormalization-group analysis 5.3.1 Flow equations 5.3.2 Basic flow properties 5.3.3 Phase diagrams 5.4 Discussion 5.5 Summary and outlook 6 Dirac from Luttinger II: Spontaneous Symmetry Breaking 6.1 Model 6.2 Phase diagram and transitions 6.3 Emergent Lorentz symmetry 6.3.1 Loop expansion near lower critical dimension 6.3.1.1 Minimal 4-Fermi model 6.3.1.2 Gross–Neveu–Heisenberg fixed point 6.3.1.3 Fate of rotational symmetry breaking 6.3.2 Loop expansion near upper critical dimension 6.3.2.1 Gross–Neveu–Yukawa–Heisenberg model 6.3.2.2 Gross–Neveu–Yukawa–Heisenberg fixed point 6.3.2.3 Fate of rotational symmetry breaking 6.4 Critical exponents 6.5 Discussion 7 Higgs Mass in Asymptotically Safe Gravity with a Dark Portal 7.1 Review: The asymptotic safety scenario for quantum gravity and matter 7.2 Review: Higgs mass, and RG flow in the SM and beyond 7.2.1 Higgs mass in the SM 7.2.2 Higgs mass bounds in bosonic portal models 7.2.3 Higgs mass in asymptotic safety 7.2.4 Higgs Portal and Asymptotic Safety 7.3 Higgs mass in an asymptotically safe dark portal model 7.3.1 The UV regime 7.3.2 Flow towards the IR 7.3.3 Infrared masses 7.3.4 From the UV to the IR – Contrasting effective field theory and asymptotic safety 7.4 Discussion 8 Conclusions Appendices A Position-space propagator for C₃-symmetric QBT B Two-sided Padé approximants for C₃-symmetric QBTs C Corrections to the mean-field nematic order-parameter effective potential due to explicit symmetry breaking D Self-energy in anisotropic Yukawa theory E Master integrals for anisotropic Yukawa theory Bibliography

In dieser Arbeit werden verschiedene Eigenschaften von dynamischen konvexen Risikomaßen für beschränkte Zufallsvariablen untersucht. Dabei gehen wir vor allem der Frage nach, wie die Risikobewertungen in verschiedenen Zeitpunkten von einander abhängen, und wie sich solche Zeitkonsistenzeigenschaften in der Dynamik der Penalty-Funktionen und Risikoprozesse widerspiegeln. Im Kapitel 2 widmen wir uns zunächst der starken Zeitkonsistenz und charakterisieren diese mithilfe von Akzeptanzmengen, Penalty-Funktionen und einer gemeinsamen Supermartingaleigenschaft des Risikoprozesses und seiner Penalty-Funktion. Die Charakterisierung durch Penalty-Funktionen liefert eine explizite Form der Doob- und der Riesz-Zerlegung des Prozesses der Penalty-Funktionen. Anschließend führen wir einen schwächeren Begriff der Zeitkonsistenz ein, den wir Besonnenheit nennen. In Analogie zu dem zeitkonsistenten Fall charakterisieren wir Besonnenheit durch Akzeptanzmengen, Penalty-Funktionen und eine bestimmte Supermartingaleigenschaft. Diese Supermartingaleigenschaft gilt allgemeiner für alle beschränkten adaptierten Prozesse, die sich ohne zusätzliches Risiko aufrechterhalten lassen. Wir nennen solche Prozesse nachhaltig und beschreiben Nachhaltigkeit durch eine gemeinsame Supermartingaleigenschaft des Prozesses und der schrittweisen Penalty-Funktionen. Dieses Resultat kann als eine verallgemeinerte optionale Zerlegung unter konvexen Restriktionen gesehen werden. Mithilfe der Supermartingaleigenschaft identifizieren wir das stark zeitkonsistente dynamische Risikomaß, das aus jedem beliebigen Risikomaß rekursiv konstruiert werden kann, als den kleinsten Prozeß, der nachhaltig ist und den Endverlust minimiert. Diese Beschreibung liefert ein neues Argument für den Einsatz von zeitkonsistenten Risikomaßen. Im Kapitel 3 diskutieren wir das asymptotische Verhalten von zeitkonsistenten und von besonnenen Risikomaßen hinsichtlich der asymptotischen Sicherheit und der asymptotischen Präzision. Im Kapitel 4 werden die allgemeinen Ergebnisse aus den Kapiteln 2 und 3 anhand des entropischen Risikomaßes und des Superhedging-Preisprozesses veranschaulicht. / In this thesis we study various properties of a dynamic convex risk measure for bounded random variables. The main subject is to investigate possible interdependence of conditional risk assessments at different times and the manifestation of these time consistency properties in the dynamics of corresponding penalty functions and risk processes. In Chapter 2 we focus first on the strong notion of time consistency and characterize it in terms of penalty functions, acceptance sets and a joint supermartingale property of the risk measure and its penalty function. The characterization in terms of penalty functions provides the explicit form of the Doob and of the Riesz decomposition of the penalty function process for a time consistent risk measure. Then we introduce and study a weaker notion of time consistency, that we call prudence. Similar to the time consistent case, we characterize prudent dynamic risk measures in terms of acceptance sets, of penalty functions and by a certain supermartingale property. This supermartingale property holds more generally for any bounded adapted process that can be upheld without any additional risk. We call such processes sustainable, and we give an equivalent characterization of sustainability in terms of a combined supermartingale property of a process and one-step penalty functions. This result can be viewed as a generalized optimal decomposition under convex constraints. The supermartingale property allows us to characterize the strongly time consistent risk measure arising from any dynamic risk measure via recursive construction as the smallest process that is sustainable and covers the final loss. Thus our discussion provides a new reason for using strongly time consistent risk measures. In Chapter 3 we discuss the limit behavior of time consistent and of prudent risk measures in terms of asymptotic safety and of asymptotic precision. In the final Chapter 4 we illustrate the general results of Chapter 2 and Chapter 3 by examples. In particular we study the entropic dynamic risk measure and the superhedging price process under convex constraints.

In dieser Doktorarbeit werden wir das Renormierungsproblem von Gravitationstheorien im Kontext der Renormierungsgruppe (RG) unter Anwendung von perturbativen und nicht-perturbativen Methoden untersuchen. Insbesondere werden wir uns auf verschiedene Gravitationsmodelle und Näherungen konzentrieren, in welchen die zentrale Rolle von einem skalaren Freiheitsgrad eingenommen wird. Wir konzentrieren uns besonders auf zwei Ansätze für Quantengravitation, die in letzter Zeit viel Aufmerksamkeit erhalten haben, nämlich den asymptotisch sicheren Fall der Gravitation und die Hořava-Lifshitz Quantengravitation. Das Prinzip der Asymptotischen Sicherheit beruht auf der Annahme, dass das hochenergetische Gravitationsregime von einem nicht-Gaußschen Fixpunkt bestimmt wird, der nicht-perturbative Renormierung und Endlichkeit der Korrelationsfunktionen sicherstellt. Wir werden die Existenz eines solchen nicht-trivialen Fixpunktes mit Hilfe der funktionalen Renormierungsgruppe untersuchen. Insbesondere werden wir den einzigen konformen Freiheitsgrad quantisieren. Die Frage nach der Existenz eines nicht-Gaußschen Fixpunktes in einem unendlich- dimensionalen Parameterraum, das heißt für eine generische f(R)-Theorie, kann jedoch nicht mit einem solchen konform reduzierten Model analysiert werden. Deshalb werden wir es untersuchen, indem wir eine skalare dynamische Äquivalentstheorie, das heißt eine generische Brans-Dicke Theorie in der lokal-Potential Näherung mit ω = 0, quantisieren. Schließlich werden wir mittels einer perturbativen RG Methode die asymptotische Freiheit der Hořava-Lifshitz Gravitationstheorie analysieren. Diese Gravitationstheorie beruht auf der Entstehung einer Anisotropie zwischen Raum und Zeit, die Newtons Konstante zu einer marginalen Koppelung werden lässt und explizit die Unitarität bewahrt. Insbesondere werden wir die Einschleifenkorrektur in 2+1 Dimensionen berechnen, indem wir nur den konformen Freiheitsgrad quantisieren. / In this Ph.D. thesis we will study the issue of renormalizability of gravitation in the context of the renormalization group (RG), employing both perturbative and non-perturbative techniques. In particular, we will focus on different gravitational models and approximations in which a central role is played by a scalar degree of freedom, since their RG flow is easier to analyze. We restrict our interest in particular to two quantum gravity approaches that have gained a lot of attention recently, namely the asymptotic safety scenario for gravity and the Hořava-Lifshitz quantum gravity. In the so-called asymptotic safety conjecture the high energy regime of gravity is controlled by a non-Gaussian fixed point which ensures non-perturbative renormalizability and finiteness of the correlation functions. We will then investigate the existence of such a non trivial fixed point using the functional renormalization group, a continuum version of the non-perturbative Wilson’s renormalization group. In particular we will quantize the sole conformal degree of freedom, which is an approximation that has been shown to lead to a qualitatively correct picture. The question of the existence of a non-Gaussian fixed point in an infinite-dimensional parameter space, that is for a generic f(R) theory, cannot however be studied using such a conformally reduced model. Hence we will study it by quantizing a dynamically equivalent scalar-tensor theory, i.e. a generic Brans-Dicke theory with ω = 0 in the local potential approximation. Finally, we will investigate, using a perturbative RG scheme, the asymptotic freedom of the Hořava-Lifshitz gravity, that is an approach based on the emergence of an anisotropy between space and time which lifts the Newton’s constant to a marginal coupling and explicitly preserves unitarity. In particular we will evaluate the one-loop correction in 2+1 dimensions quantizing only the conformal degree of freedom.